Methods and systems for maximum-likelihood detection using post-squaring compensation

ABSTRACT

A “post-squaring” detection algorithm, and related devices, that may reduce the complexity of maximum likelihood detection (MLD) schemes while preserving their performance is provided. Rather than search for optimum metrics (such as minimum distance metrics) based on squared norm values, a search may be based on un-squared norm metrics, and the squaring may be postponed, for example, until subsequent log-likelihood ratio (LLR) computation. For certain embodiments, approximations of un-squared norm values may significantly reduce computation complexity.

TECHNICAL FIELD

The present disclosure generally relates to MIMO OFDM communicationsystems and more specifically to a method to reduce computationalcomplexity of Max-Log-MAP based maximum-likelihood (ML) detectionapplied at the receiver.

BACKGROUND

A multiple-input multiple-output (MIMO) communication system employsmultiple (N_(T)) transmit antennas and multiple (N_(R)) receive antennasfor data transmission. A MIMO channel formed by the N_(T) transmit andN_(R) receive antennas may be decomposed into N_(S) independentchannels, with N_(S)≦min {N_(T), N_(R)}. Each of the N_(S) independentchannels is also referred to as a spatial sub-channel of the MIMOchannel and corresponds to a dimension. The MIMO system can provideimproved performance (e.g., increased transmission capacity) over thatof a single-input single-output (SISO) communication system if theadditional dimensionalities created by the multiple transmit and receiveantennas are utilized.

A wideband MIMO system typically experiences frequency selective fading,i.e. different amounts of attenuation across the system bandwidth. Thisfrequency selective fading causes inter-symbol interference (ISI), whichis a phenomenon whereby each symbol in a received signal acts asdistortion to subsequent symbols in the received signal. This distortiondegrades performance by impacting the ability to correctly detect thereceived symbols. As such, ISI is a non-negligible noise component thatmay have a large impact on the overall signal-to-noise-and-interferenceratio (SNR) for systems designed to operate at high SNR levels, such asMIMO systems. In such systems, equalization may be used at the receiversto combat ISI. However, the computational complexity required to performequalization is typically significant or prohibitive for mostapplications.

Orthogonal frequency division multiplexing (OFDM) may be used to combatISI without the use of computationally intensive equalization. An OFDMsystem effectively partitions the system bandwidth into a number of(N_(F)) frequency sub-channels, which may be referred to as sub-bands orfrequency bins. Each frequency sub-channel is associated with arespective subcarrier frequency upon which data may be modulated. Thefrequency sub-channels of the OFDM system may experience frequencyselective fading (i.e., different amounts of attenuation for differentfrequency sub-channels) depending on the characteristics (e.g.,multipath profile) of the propagation path between transmit and receiveantennas. With OFDM, the ISI due to the frequency selective fading maybe combated by repeating a portion of each OFDM symbol (i.e., appendinga cyclic prefix to each OFDM symbol), as is known in the art. A MIMOsystem may thus advantageously employ OFDM to combat ISI.

In order to increase the transmission data rate and spectral efficiencyof the system, spatial multiplexing may be applied at the transmitterwhere different and independent data streams may be communicated over aplurality of spatial sub-channels. The detection accuracy of thereceiver can be severely degraded due to a strong multiple accessinterference (interference of data streams transmitted from differentantennas). Moreover, spatial and frequency sub-channels may experiencedifferent channel conditions (e.g., fading and multipath effects) andmay achieve different SNRs. Also, channel conditions may vary over time.

Different techniques can be applied at the receiver to accurately detectinformation data transmitted from a plurality of antennas over spatialand frequency sub-channels. Suppression of the multiple accessinterference in MIMO-OFDM system can be achieved by applying maximumlikelihood (ML) detection at the receiver.

The ML detection is optimal algorithm in terms of accuracy becauselikelihoods of all symbol vectors that can be possibly transmitted froma plurality of antennas may be evaluated. On the other side,computational complexity may be large especially in systems with highspectral efficiency that employ large number of transmit antennas andhigh-order modulation types.

The Log-MAP (maximum a posteriori) ML detection can be considered as anideal maximum-likelihood algorithm in terms of error rate performance.It utilizes squared l₂ norm as a metric for calculation oflog-likelihood ratios (hereinafter abbreviated as LLRs) of codedtransmission bits. Complexity of direct computation of squared l₂ normsmay be large, especially for high order modulation types and/or highdimensional Multiple-Input Multiple-Output (MIMO) wireless systems. Dueto a high computational complexity of the Log-MAP ML algorithm, theMax-Log-MAP approach for ML detection may be preferred for actualhardware implementation where summation operations can be replaced withcomparison operations. But, this approach also requires computation ofsquared l₂ norms.

Therefore, there is a need in the art to reduce computational complexityof the Max-Log-MAP based ML detection in systems with high spectralefficiency.

SUMMARY

Certain embodiments of the present disclosure provide a method fordetection of data transmitted on a plurality of spatial channels in amultiple-input multiple-output wireless communications system. Themethod generally includes calculating un-squared norm metrics for a setof hypothesized symbols that that correspond to coded bits transmittedover the spatial channels, searching the un-squared norm metrics foroptimum metrics, squaring optimum metrics found during the searching toobtain post-squared metrics, and calculating log-likelihood ratio valuesfor coded bits transmitted over the spatial channels using post-squaredmetrics.

Certain embodiments of the present disclosure provide an apparatus fordetection of data transmitted on a plurality of spatial channels in amultiple-input multiple-output wireless communications system. Theapparatus generally includes logic for calculating un-squared normmetrics for a set of hypothesized symbols that that correspond to codedbits transmitted over the spatial channels, logic for searching theun-squared norm metrics for optimum metrics, logic for squaring optimummetrics found during the searching to obtain post-squared metrics, andlogic for calculating log-likelihood ratio values for coded bitstransmitted over the spatial channels using post-squared metrics.

Certain embodiments of the present disclosure provide an apparatus fordetection of data transmitted on a plurality of spatial channels in amultiple-input multiple-output wireless communications system. Theapparatus generally includes means for calculating un-squared normmetrics for a set of hypothesized symbols that that correspond to codedbits transmitted over the spatial channels, means for searching theun-squared norm metrics for optimum metrics, means for squaring optimummetrics found during the searching to obtain post-squared metrics, andmeans for calculating log-likelihood ratio values for coded bitstransmitted over the spatial channels using post-squared metrics.

Certain embodiments of the present disclosure provide a computer-programproduct for detection of data transmitted on a plurality of spatialchannels in a multiple-input multiple-output wireless communicationssystem, comprising a computer readable medium having instructions storedthereon, the instructions being executable by one or more processors.The instructions generally include instructions for calculatingun-squared norm metrics for a set of hypothesized symbols that thatcorrespond to coded bits transmitted over the spatial channels,instructions for searching the un-squared norm metrics for optimummetrics, instructions for squaring optimum metrics found during thesearching to obtain post-squared metrics, and instructions forcalculating log-likelihood ratio values for coded bits transmitted overthe spatial channels using post-squared metrics.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above recited features of the presentdisclosure can be understood in detail, a more particular description,briefly summarized above, may be had by reference to embodiments, someof which are illustrated in the appended drawings. It is to be noted,however, that the appended drawings illustrate only certain typicalembodiments of this disclosure and are therefore not to be consideredlimiting of its scope, for the description may admit to other equallyeffective embodiments.

FIG. 1 illustrates an example wireless communication system, inaccordance with certain embodiments of the present disclosure.

FIG. 2 illustrates various components that may be utilized in a wirelessdevice in accordance with certain embodiments of the present disclosure.

FIG. 3 illustrates an example transmitter and an example receiver thatmay be used within a wireless communication system in accordance withcertain embodiments of the present disclosure.

FIG. 4 illustrates a block diagram of a generic MIMO-OFDM wirelesssystem in accordance with certain embodiments of the present disclosure.

FIG. 5 shows a block diagram of a typical implementation of Max-Log-MAPML detection in accordance with certain embodiments of the presentdisclosure.

FIG. 6 shows a block diagram of Max-Log-MAP QRML detector in accordancewith certain embodiments of the present disclosure.

FIG. 7 shows a process of Max-Log-MAP ML detection with post-squaringcompensation in accordance with certain embodiments of the presentdisclosure.

FIG. 7A illustrates example components capable of performing theoperations illustrated in FIG. 7.

FIG. 8 shows a block diagram of Max-Log-MAP ML detector withpost-squaring compensation in accordance with certain embodiments of thepresent disclosure.

FIG. 9 shows a process of Max-Log-MAP QRML detection with post-squaringcompensation in accordance with certain embodiments of the presentdisclosure.

FIG. 9A illustrates example components capable of performing theoperations illustrated in FIG. 9.

FIG. 10 shows a block diagram of Max-Log-MAP QRML detector withpost-squaring compensation in accordance with certain embodiments of thepresent disclosure.

FIG. 11 shows a comparison of computational complexity for several MLdetection schemes used in the simulations in accordance with certainembodiments of the present disclosure.

FIG. 12 shows a summary of ML detection schemes used in the simulationsin accordance with certain embodiments of the present disclosure.

FIG. 13 shows a packet error rate performance loss in dB units ofdifferent ML algorithms relative to the QRML detection with metric basedon squared l₂ norm.

DETAILED DESCRIPTION

For certain embodiments of the present disclosure, a post squaredcompensation of un-squared l₂ norms is proposed as an alternative methodfor calculation of LLRs. Computational complexity of this scheme issignificantly smaller than complexity of conventional Max-Log-MAP basedML algorithm, while the accuracy is preserved.

The proposed algorithm may be referred to as the post-squaringcompensation maximum-likelihood detection (hereinafter abbreviated asPC-ML detection). Instead of direct calculation of squared l₂ normsemployed in the conventional Max-Log-MAP based ML algorithm, non-squaredl₂ norms may now be utilized. Squaring of l₂ norms may be postponeduntil the search for minima non-squared metrics is finished.

This approach may provide identical final result as a direct calculationof squared metrics performed before searching for minima values.Therefore, the PC-ML scheme achieves the same detection accuracy as theconventional Max-Log-MAP based ML algorithm. It can be shown that thecomputational complexity of the PC-ML detection algorithm is smallerthan that of conventional Max-Log-MAP based ML detection sincecomputation of non-squared l₂ norms can be accurately approximated withcomparison and addition operations instead of multiplications that areutilized for direct computation of squared l₂ norms. Furthermore,because only minima metrics are post-squared, overhead in thecomputational complexity of the PC-ML algorithm introduced by thepost-squaring compensation is small, and it does not affect the overallcomputational complexity.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments.

Exemplary Wireless Communication System

The techniques described herein may be used for various broadbandwireless communication systems, including communication systems that arebased on an orthogonal multiplexing scheme. Examples of suchcommunication systems include Orthogonal Frequency Division MultipleAccess (OFDMA) systems, Single-Carrier Frequency Division MultipleAccess (SC-FDMA) systems, and so forth. An OFDMA system utilizesorthogonal frequency division multiplexing (OFDM), which is a modulationtechnique that partitions the overall system bandwidth into multipleorthogonal sub-carriers. These sub-carriers may also be called tones,bins, etc. With OFDM, each sub-carrier may be independently modulatedwith data. An SC-FDMA system may utilize interleaved FDMA (IFDMA) totransmit on sub-carriers that are distributed across the systembandwidth, localized FDMA (LFDMA) to transmit on a block of adjacentsub-carriers, or enhanced FDMA (EFDMA) to transmit on multiple blocks ofadjacent sub-carriers. In general, modulation symbols are sent in thefrequency domain with OFDM and in the time domain with SC-FDMA.

One specific example of a communication system based on an orthogonalmultiplexing scheme is a WiMAX system. WiMAX, which stands for theWorldwide Interoperability for Microwave Access, is a standards-basedbroadband wireless technology that provides high-throughput broadbandconnections over long distances. There are two main applications ofWiMAX today: fixed WiMAX and mobile WiMAX. Fixed WiMAX applications arepoint-to-multipoint, enabling broadband access to homes and businesses,for example. Mobile WiMAX offers the full mobility of cellular networksat broadband speeds.

IEEE 802.16x is an emerging standard organization to define an airinterface for fixed and mobile broadband wireless access (BWA) systems.These standards define at least four different physical layers (PHYs)and one media access control (MAC) layer. The OFDM and OFDMA physicallayer of the four physical layers are the most popular in the fixed andmobile BWA areas respectively.

FIG. 1 illustrates an example of a wireless communication system 100 inwhich embodiments of the present disclosure may be employed. Thewireless communication system 100 may be a broadband wirelesscommunication system. The wireless communication system 100 may providecommunication for a number of cells 102, each of which is serviced by abase station 104. A base station 104 may be a fixed station thatcommunicates with user terminals 106. The base station 104 mayalternatively be referred to as an access point, a Node B or some otherterminology.

FIG. 1 depicts various user terminals 106 dispersed throughout thesystem 100. The user terminals 106 may be fixed (i.e., stationary) ormobile. The user terminals 106 may alternatively be referred to asremote stations, access terminals, terminals, subscriber units, mobilestations, stations, user equipment, etc. The user terminals 106 may bewireless devices, such as cellular phones, personal digital assistants(PDAs), handheld devices, wireless modems, laptop computers, personalcomputers, etc.

A variety of algorithms and methods may be used for transmissions in thewireless communication system 100 between the base stations 104 and theuser terminals 106. For example, signals may be sent and receivedbetween the base stations 104 and the user terminals 106 in accordancewith OFDM/OFDMA techniques. If this is the case, the wirelesscommunication system 100 may be referred to as an OFDM/OFDMA system.

A communication link that facilitates transmission from a base station104 to a user terminal 106 may be referred to as a downlink (DL) 108,and a communication link that facilitates transmission from a userterminal 106 to a base station 104 may be referred to as an uplink (UL)110. Alternatively, a downlink 108 may be referred to as a forward linkor a forward channel, and an uplink 110 may be referred to as a reverselink or a reverse channel.

A cell 102 may be divided into multiple sectors 112. A sector 112 is aphysical coverage area within a cell 102. Base stations 104 within awireless communication system 100 may utilize antennas that concentratethe flow of power within a particular sector 112 of the cell 102. Suchantennas may be referred to as directional antennas.

FIG. 2 illustrates various components that may be utilized in a wirelessdevice 202 that may be employed within the wireless communication system100. The wireless device 202 is an example of a device that may beconfigured to implement the various methods described herein. Thewireless device 202 may be a base station 104 or a user terminal 106.

The wireless device 202 may include a processor 204 which controlsoperation of the wireless device 202. The processor 204 may also bereferred to as a central processing unit (CPU). Memory 206, which mayinclude both read-only memory (ROM) and random access memory (RAM),provides instructions and data to the processor 204. A portion of thememory 206 may also include non-volatile random access memory (NVRAM).The processor 204 typically performs logical and arithmetic operationsbased on program instructions stored within the memory 206. Theinstructions in the memory 206 may be executable to implement themethods described herein.

The wireless device 202 may also include a housing 208 that may includea transmitter 210 and a receiver 212 to allow transmission and receptionof data between the wireless device 202 and a remote location. Thetransmitter 210 and receiver 212 may be combined into a transceiver 214.A plurality of transmit antennas 216 may be attached to the housing 208and electrically coupled to the transceiver 214. The wireless device 202may also include (not shown) multiple transmitters, multiple receivers,and multiple transceivers.

The wireless device 202 may also include a signal detector 218 that maybe used in an effort to detect and quantify the level of signalsreceived by the transceiver 214. The signal detector 218 may detect suchsignals as total energy, energy per subcarrier per symbol, powerspectral density and other signals. The wireless device 202 may alsoinclude a digital signal processor (DSP) 220 for use in processingsignals.

The various components of the wireless device 202 may be coupledtogether by a bus system 222, which may include a power bus, a controlsignal bus, and a status signal bus in addition to a data bus.

FIG. 3 illustrates an example of a transmitter 302 that may be usedwithin a wireless communication system 100 that utilizes OFDM/OFDMA.Portions of the transmitter 302 may be implemented in the transmitter210 of a wireless device 202. The transmitter 302 may be implemented ina base station 104 for transmitting data 306 to a user terminal 106 on adownlink 108. The transmitter 302 may also be implemented in a userterminal 106 for transmitting data 306 to a base station 104 on anuplink 110.

Data 306 to be transmitted is shown being provided as input to MIMOencoder 312. The MIMO encoder may encode the data 306 and map them ontoM constellation points. The mapping may be done using some modulationconstellation, such as binary phase-shift keying (BPSK), quadraturephase-shift keying (QPSK), 8 phase-shift keying (8PSK), quadratureamplitude modulation (QAM), etc, or null (zero valued) modulation if thesub-carrier is not assigned. Thus, the MIMO encoder 312 may outputN_(FFT) parallel symbol streams 316 per antenna path, each symbol stream316 corresponding to N_(FFT) orthogonal subcarriers of the inverse fastFourier transforms (IFFTs) 320. These N_(FFT) parallel symbol streams316 are represented in the frequency domain and may be converted intoN_(FFT) parallel time domain sample streams 318 by IFFT components 320.

A brief note about terminology will now be provided. N_(T)×N_(FFT)parallel modulations in the frequency domain are equal to N_(T)×N_(FFT)modulation symbols in the frequency domain, which are equal to N_(T)parallel N_(FFT) mapping and N_(T) parallel N_(FFT)-point IFFTs in thefrequency domain, which is equal to N_(T) (useful) OFDM symbols in thetime domain, each of which is equal to N_(FFT) samples in the timedomain, where N_(T) is the number of transmit antennas 330. Afterinserting Guard samples, one OFDM symbol in the time domain, N_(S), isequal to N_(CP) (the number of guard samples per OFDM symbol)+N_(FFT)(the number of useful samples per OFDM symbol) samples.

For one of N_(T) antenna path, the N_(FFT) parallel time domain samplestreams 318 may be converted into an OFDM/OFDMA symbol stream 322 byparallel-to-serial (P/S) converter 324. Guard insertion component 326may insert guard interval between successive OFDM/OFDMA symbols in theOFDM/OFDMA symbol stream 322 generating N_(S)=N_(CP)+N_(FFT) samples.The signal from the guard insertion component 326 may then beup-converted to a desired transmit frequency band by a radio frequency(RF) front end component 328, and the antenna array 330 may thentransmit the resulting signal 332 across multiple spatial sub-channels334.

FIG. 3 also illustrates an example of a receiver 304 that may be usedwithin a wireless device 202 that utilizes OFDM/OFDMA. Portions of thereceiver 304 may be implemented in the receiver 212 of a wireless device202. The receiver 304 may be implemented in a user terminal 106 forreceiving data 306 from a base station 104 on a downlink 108. Thereceiver 304 may also be implemented in a base station 104 for receivingdata 306 from a user terminal 106 on an uplink 110.

The transmitted signal 332 is shown traveling over a plurality ofspatial sub-channels 334. When a signal 332′ is received by the antennaarray 330′, the received signal 332′ may be downconverted to a basebandsignal by RF front end components 328′. Guard removal components 326′may then remove the guard intervals that were inserted betweenOFDM/OFDMA symbols by the guard insertion components 326.

For one of N_(T) antenna path, the output of the guard removal component326′ may be provided to S/P converter 324′. The S/P converter 324′ maydivide the OFDM/OFDMA symbol stream 322′ into the N_(FFT) paralleltime-domain symbol streams 318′, each of which corresponds to one of theN_(FFT) orthogonal subcarriers. Fast Fourier transform (FFT) component320′ may convert the N_(FFT) parallel time-domain symbol streams 318′into the frequency domain and output N_(FFT) parallel frequency-domainsymbol streams 316′.

A MIMO decoder 312′ may perform the inverse of the encoding and symbolmapping operations that were performed by the MIMO encoder 312 therebyoutputting the same number of data stream 306′ as the data 306. Ideally,this data stream 306′ corresponds to the data 306 that was provided asinput to the transmitter 302. Note that elements 312′, 316′, 320′, 318′and 324′ may all be found on a in a baseband processor 340′.

Exemplary MIMO-OFDM System Model

FIG. 4 shows a block diagram of generic multiple-input multiple-output(MIMO) OFDM wireless communication system with N_(T) transmit and N_(R)receive antennas. The system model for the k^(th) sub-carrier (frequencysub-channel) may be represented with linear equation:

y _(k) =H _(k) X _(k) +n _(k) , k=1,2, . . . , N _(FFT)   (1)

where N_(FFT) is the number of orthogonal sub-carriers (frequency bins)in MIMO-OFDM system.

In equations and accompanying disclosure below, the sub-carrier index kis omitted for simplicity. Therefore, the system model can be re-writtenin the simple notation as:

$\begin{matrix}{y = {{Hx} + n}} & (2) \\{y = \left\lbrack {y_{1}\mspace{14mu} y_{2}\mspace{14mu} \ldots \mspace{14mu} y_{N_{r}}} \right\rbrack^{T}} & (3) \\{H = {\left\lbrack {h_{1}\mspace{14mu} h_{2}\mspace{14mu} \ldots \mspace{14mu} h_{N_{t}}} \right\rbrack = \begin{bmatrix}h_{11} & h_{12} & \ldots & h_{1\; N_{t}} \\\; & \ldots & \; & \; \\h_{N_{r}1} & h_{N_{r}2} & \ldots & h_{N_{r}N_{t}}\end{bmatrix}}} & (4) \\{x = \left\lbrack {x_{1}\mspace{14mu} x_{2}\mspace{14mu} \ldots \mspace{14mu} x_{N_{t}}} \right\rbrack^{T}} & (5) \\{n = \left\lbrack {n_{1}\mspace{14mu} n_{2}\mspace{14mu} \ldots \mspace{14mu} n_{N_{r}}} \right\rbrack^{T}} & (6)\end{matrix}$

where y is [N_(r)×1] received symbol vector, H is [N_(r)×N_(t)] channelmatrix and h_(j) is its j^(th) column vector that contains channel gainsbetween the transmit antenna j and all N_(r) receive antennas, x is[N_(t)×1] transmitted symbol vector, n is [N_(r)×1] complex noise vectorwith covariance matrix E(nn^(H)).

As illustrated in FIG. 4, the transmission signal may be first encodedby MIMO encoder 410. A redundancy may be included to protect theinformation data during the transmission over noisy wireless channels.An encoded signal may then be split into N_(t) spatial data streams x₁,x₂, . . . , x_(Nt), as shown in FIG. 4. A plurality of spatial datastreams can be converted into a time domain by utilizing Inverse FastFourier Transform (IFFT) units 412 ₁, . . . , 412 _(Nt). The signal maythen be up converted to a desired transmission frequency band andtransmitted from N_(t) transmit antennas 414 ₁, . . . , 414 _(Nt) overN_(r)·N_(t) single-input single-output (SISO) channels.

N_(r) receive antennas 416 ₁, . . . , 416 _(Nr) are employed at thereceiver. Received data streams can be converted back into a frequencydomain by using the Fast Fourier Transform (FFT) units 418 ₁, . . . ,418 _(Nr). A frequency domain signal may be input into a MIMO detector420 that generates reliability messages for coded bits transmitted overa plurality of spatial sub-channels. A reliability message represents aprobability that the particular transmitted coded bit is either bit “0”or bit “1”. This information can be passed to the outer MIMO channeldecoder 422, and the estimated information data {circumflex over (x)}for a plurality of spatial sub-channels (transmit antennas) areavailable after removing the redundancy included at the transmitter.

For a given vector v, l_(p) norm can be defined as:

$\begin{matrix}{v = \left\lbrack {v_{1}\mspace{14mu} v_{2}\mspace{14mu} \ldots \mspace{14mu} v_{j}\mspace{14mu} \ldots \mspace{14mu} v_{N}} \right\rbrack^{T}} & (7) \\\begin{matrix}{l_{p} = {v}_{p}} \\{{= \left( {{v_{1}}^{p} + {{v_{2}}^{p}\mspace{14mu} \ldots} + {v_{N}}^{p}} \right)^{\frac{1}{p}}}\mspace{14mu}} \\{= \left( {\sum\limits_{j = 1}^{N}{v_{j}}^{p}} \right)^{\frac{1}{p}}}\end{matrix} & (8)\end{matrix}$

where p is a real number with p≧1, and the scalar element v_(j) can beeither real or complex number.

The squared l_(p) norm may be defined as follows:

Squared l_(p)=sl_(p)=l_(p) ²=∥v∥_(p) ²   (9)

Exemplary Maximum-Likelihood Detection

The maximum likelihood (ML) detection is well-known technique in the artwhich utilizes maximum a posteriori (MAP) or equivalent Log-MAPalgorithm to find the most likely transmitted modulation symbols. The MLdetection achieves optimal accuracy because it evaluates all modulationsymbols that can be potentially transmitted. The Log-MAP detector useslog likelihood ratios (LLRs) of coded bits to decide whether a bit “0”or a bit “1” is communicated over a wireless channel.

If b_(k) is the k^(th) bit of the transmitted symbol vector x, the bitLLR L (b_(k)) can be represented as follows:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {{LLR}\left( b_{k} \middle| y \right)}} \\{= {\log \left\lbrack \frac{P\left( {b_{k} = \left. 0 \middle| y \right.} \right)}{P\left( {b_{k} = \left. 1 \middle| y \right.} \right)} \right\rbrack}} \\{= {{\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{P\left( x \middle| y \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{P\left( x \middle| y \right)}} \right\rbrack} = {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{{p\left( y \middle| x \right)}{P(x)}}}{\sum\limits_{{x\text{:}b_{k}} = 1}{{p\left( y \middle| x \right)}{P(x)}}} \right\rbrack}}} \\{= {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{p\left( y \middle| x \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{p\left( y \middle| x \right)}} \right\rbrack}}\end{matrix} & (10)\end{matrix}$

where expression “x:b_(k)=0” denotes a set of candidate transmissionbits x with the k^(th) information bit equal to “0”, expression“x:b_(k)=1” denotes a set of candidate transmission bits x with thek^(th) information bit equal to “1”, p(x) is a probability densityfunction of candidate vector x, P(x) is a probability of x, and it isassumed that x is equally distributed.

The Gaussian probability density function may be associated with thetransmission symbol vector x. In this case, expression (10) may besimplified as:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {{LLR}\left( b_{k} \middle| y \right)}} \\{= {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{p\left( y \middle| x \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{p\left( y \middle| x \right)}} \right\rbrack}} \\{= {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{\exp \; \left( {- {d(x)}} \right)}} \right\rbrack}}\end{matrix} & (11)\end{matrix}$

where metric d(x) may be defined as:

$\begin{matrix}\begin{matrix}{{d(x)} = {d\left( {x_{1},{\ldots \mspace{14mu} x_{j}\mspace{14mu} \ldots}\mspace{14mu},x_{N_{t}}} \right)}} \\{= \frac{{{y - {Hx}}}^{2}}{\sigma_{n}^{2}}}\end{matrix} & (12)\end{matrix}$

For j^(th) spatial data stream, x_(j) ε C^(M=2) ^(R) where M is thenumber of constellation points and B is a modulation order (number ofbits per modulation symbol). The operational complexity of this searchalgorithm is therefore proportional to O(M^(N) ^(t) ) and corresponds tothe number of hypothesized transmission symbol vectors x in equation(11).

In order to reduce computational complexity of the Log-MAP approach, theMax-Log-MAP algorithm may be applied. If the Gaussian probabilitydensity function for the transmission symbol vector x is again assumed,the LLR for the k^(th) bit of the transmission signal vector x L (b_(k))may be computed as:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {{LLR}\left( b_{k} \middle| y \right)}} \\{= {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{p\left( y \middle| x \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{p\left( y \middle| x \right)}} \right\rbrack}} \\{\approx {\log \left\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}{p\left( y \middle| x \right)}}{\max\limits_{{x\text{:}b_{k}} = 1}{p\left( y \middle| x \right)}} \right\rbrack}} \\{= {\log \left\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\max\limits_{{x\text{:}b_{k}} = 1}{\exp \left( {- {d(x)}} \right)}} \right\rbrack}} \\{= {{\min\limits_{{x\text{:}b_{k}} = 1}{d(x)}} - {\min\limits_{{x\text{:}b_{k}} = 0}{d(x)}}}}\end{matrix} & (13)\end{matrix}$

The operational complexity of this approach is still proportional toO(M^(N) ^(t) ), but, as shown in expression (13), the search operationsmay now be used instead of summation operations of the Log-MAP approachgiven by equation (11).

As shown by equation (12), calculation of LLRs in both algorithms may bebased on l₂ ² norms. Assuming unitary variance of effective noise at thereceiver (after pre-whitening, for example), the c^(th) metric d_(c)from equation (12) and (13) may be represented as:

d_(c)=l₂ ²=∥v∥₂ ²   (14)

-   -   where, v=y−Hx, c=1,2, . . . , M^(N) ^(t)

FIG. 5 shows a block diagram of a typical implementation of theMax-Log-MAP ML detection. Channel estimates of spatial sub-channels forone frequency sub-band in the form of matrix H and vector of receivedsamples y can be input into unit 510. All possible M^(N) ^(T) vectorsymbols x that can be transmitted from N_(T) antennas may behypothesized and M^(N) ^(T) l₂ ² norms may be calculated as specified byequation (14). Following this, unit 520 may perform search for minimametrics for every transmission bit k=1, 2, . . . , N_(T)·B for allhypotheses x for which transmission bit k is equal to bit “0”, and forall hypotheses x for which transmission bit k is equal to bit “1”.Therefore, computational complexity of this search process is againproportional to O(N_(T)·B·M^(N) ^(T) ).

Based on found minima metrics for every transmission bit k=1, 2, . . . ,N_(T)·B, bit LLRs may be calculated in unit 530 as shown in equation(13). Calculated LLRs for all N_(T)·B coded bits transmitted over aplurality of spatial sub-channels on one frequency bin may then bepassed to the outer channel decoder 540 that provides decoded spatialdata streams at its output.

Exemplary Maximum-Likelihood Detection based on QR Decomposition

The Max-Log-MAP maximum-likelihood detection may be also implementedwith the pre-processing based on QR decomposition (hereinafterabbreviated as QRML detection). By applying the pre-processing with QRdecomposition, the computational complexity of Max-Log-MAP based MLdetection can be reduced, while preserving detection accuracy.

In the QRML detection, instead of all N_(T) spatial data streams,(N_(T)−1) streams may be hypothesized. A remaining spatial data streamcan be deterministic, and may be generated by rotating received symbolswith a unitary matrix obtained after QR decomposition of the channelmatrix. Filtered symbols may then be sliced to obtain estimatedmodulation symbols for that particular spatial data stream. A process ofhypothesizing (N_(T)−1) spatial data streams and filtering a remainingstream may be performed for every spatial data stream. Because not allpossible spatial streams are hypothesized, computational complexity ofthe candidate search process may be reduced compare to conventionalmaximum-likelihood detection.

FIG. 6 shows a block diagram of the Max-Log-MAP QRML detector. Thechannel matrix H may be permuted in unit 610 for every spatial datastreams j=1, 2, . . . , N_(T) such that the rightmost column of thepermuted matrix H_(P) corresponds to the j^(th) decoded spatial datastream:

H_(P)=└h_(p(1)) h_(p(2)) . . . h_(p(N) _(t) ₁₎ h_(p(N) _(t)_()=j)┘  (15)

The QR decomposition of the permuted channel matrix for the j^(th)spatial data stream may be represented as:

H_(P)=QR   (16)

The QR decomposition operation may be performed for every spatial datastream in the system by utilizing the appropriate permuted channelmatrix.

The rotation (or, equivalently, the zero-forcing filtering) of thereceived symbol vector for every spatial data stream may also beperformed in unit 610 by utilizing the appropriate unitary matrix Q fromthe QR decomposition operation given by equation (16):

y _(r) =Q ^(H) ·y   (17)

For every spatial data stream j=1, 2, . . . , N_(T), the rotatedreceived symbol vector y_(r) may be sliced by unit 620 to obtainestimated modulation symbols for that particular spatial data stream j.All other (N_(T)−1) spatial streams may be hypothesized with allpossible transmission symbols. Therefore, unit 620 may calculateN_(T)·M^(N) ^(r) ⁻¹ metrics based on l₂ ² norms as:

d_(r)=l₂ ²=∥v∥₂ ²   (18)

-   -   where, v=y_(r)−Rx, c=1, 2, . . . , N_(r)M^(N) ^(t) ⁻¹        where y_(r) is a rotated received vector from equation (17), and        matrix R represents an upper triangular matrix obtained from        equation (16) after QR decomposition of the appropriately        permuted channel matrix given by equation (15) for the        particular spatial data stream j.

Unit 630 performs search for minima metrics for every transmission bitk=1, 2, . . . , N_(T)·B assuming all hypotheses x for which bit k isequal to bit “0”, and all hypotheses x for which bit k is equal to bit“1”. Computational complexity of the candidate search process istherefore proportional to O(N_(T)·B·N_(T)·M^(N) ^(r) ⁻¹) since there areN_(T)·M^(N) ^(T) ⁻¹ hypotheses to be compared for every coded bit.

Based on found minima metrics for every bit k=1, 2, . . . , N_(T)·B thatmay be transmitted on a single frequency bin, LLRs can be calculated inunit 640 by subtracting minima metrics:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {{LLR}\left( b_{k} \middle| y \right)}} \\{= {\log\left\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\sum\limits_{{x\text{:}b_{k}} = 1}{\exp \; \left( {- {d(x)}} \right)}} \right\rbrack}} \\{\approx {\log \left\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\max\limits_{{x\text{:}b_{k}} = 1}{\exp \left( {- {d(x)}} \right)}} \right\rbrack}} \\{= {{\min\limits_{{x\text{:}b_{k}} = 1}{d(x)}} - {\min\limits_{{x\text{:}b_{k}} = 0}{d(x)}}}}\end{matrix} & (19) \\{where} & \; \\\begin{matrix}{{{d(x)} = {d\left( {x_{1},{\ldots \mspace{14mu} x_{j}\mspace{14mu} \ldots}\mspace{14mu},x_{N_{t}}} \right)}}\mspace{14mu}} \\{= \frac{{{y_{r} - {Rx}}}^{2}}{\sigma_{n}^{2}}}\end{matrix} & (20)\end{matrix}$

It can be observed from equation (20) that a metric for LLR calculationis still l₂ ² norm. Assuming unitary variance of effective noise at thereceiver (for example, after pre-whitening), the c^(th) metric d_(c)from metric calculation unit 620 can be obtained as:

d_(c)=l₂ ²=∥v∥₂ ²   (21)

-   -   where, v=y_(r)−Rx, c=1, 2, . . . , N_(t)M^(N) ^(t) ⁻¹

Computed LLRs for all N_(T)·B coded bits transmitted over a plurality ofspatial sub-channels using a single frequency bin may be then passed tothe outer channel decoder 650 that provides decoded spatial data streamsat the output.

The QRML detection is an optimal solution referencing Max-Log-MAPalgorithm, and thus provides identical error-rate performance as theMax-Log-MAP based ML detection. This may be confirmed by simulationresults, which are also provided in the present disclosure.Computational complexity of the candidate search process is reduced.

Exemplary Maximum-Likelihood Detection with Post-Squaring Composition

The Max-Log-MAP ML detection with post-squaring compensation of norms(PC-ML detection) may be proposed to reduce computational complexity ofthe Max-Log-MAP ML algorithm while preserving the error-rateperformance. In the PC-ML detection, the transmission signal may behypothesized based on un-squared l₂ norms instead of squared l₂ norms,and then searching for minima un-squared distances may be performed.Squaring of metrics may be postponed until minima un-squared norms aredetermined. By computing un-squared norms and by post-squaring only alimited number of calculated metrics, computational complexity may besignificantly reduced, while identical detection accuracy can beachieved as for previously presented ML and QRML solutions.

FIG. 7 shows a process of Max-Log-MAP ML detection with post-squaringcompensation, and FIG. 8 illustrates an example block diagram ofMax-Log-MAP ML detector with post-squaring compensation that performsprocessing flow from FIG. 7. Initially, M^(N) ^(T) transmission symbolcandidates may be hypothesized based on the received signal vector andchannel estimates, at 710. Metrics may be calculated based on un-squaredl₂ norms instead of l₂ ² norms, as it is also illustrated with unit 810in FIG. 8. At 720, for every transmission bit k=1, 2, . . . , N_(T)·B,unit 820 performs search for minima metrics for all hypotheses x forwhich bit k is equal to bit “0”, and for all hypotheses x for which bitk is equal to bit “1”. The computational complexity of the searchprocess is therefore proportional to O(N_(T)·B·M^(N) ^(T) ).

Once the minima metrics for all coded bits are determined, unit 830 inFIG. 8 may perform the post-squaring compensation of the minima metrics,at 730. The post-squared minima metrics of the PC-ML detection producesidentical results as minima metrics of the original ML detection, i.e.l₂ ² norms are equivalent to the post-squared l₂ norms.

From equation (14), the relationship between l₂ ² norm and l₂ norm maybe obtained as follows:

$\begin{matrix}{\begin{matrix}{d_{c} = {l_{2}^{2} = {v}_{2}^{2}}} \\{= \left( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} \right)^{2}} \\{= {\left( d_{l_{2},c} \right)^{2} = \left( l_{2} \right)^{2}}}\end{matrix}{{where},{v = {y - {Hx}}},{c = 1},2,\ldots \mspace{11mu},M^{N_{t}}}} & (22)\end{matrix}$

The post squaring of minima metric may be applied to recover squared l₂norms required for bit LLR calculation. Let d_(l) ₂ _(,min) be thearbitrary minimum l₂ norm as a result of minima search:

d _(l) ₂ _(,min)=min(d _(l) ₂ ^(,c))   (23)

The post-squaring of the smallest l₂ norm results into equivalentminimum l₂ ² norm:

$\begin{matrix}\begin{matrix}{d_{\min} = \left( d_{l_{2},\min} \right)^{2}} \\{= \left( {\min\limits_{x}\left( {v}_{2} \right)} \right)^{2}} \\{= \left( {\min\limits_{x}\left( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} \right)} \right)^{2}} \\{= {\min\limits_{x}\left( \left( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} \right)^{2} \right)}} \\{= {\min\limits_{x}\left( {v}_{2}^{2} \right)}}\end{matrix} & (24)\end{matrix}$

Therefore, if there is no approximation for calculating l₂ norms, thenthe result of minima search algorithm using l₂ norms may be equivalentto the result of minima search using l₂ ² norms.

At 740, for every bit k=1, 2, . . . , N_(T)·B transmitted on a singlefrequency bin, bit LLR L (b_(k)) may be calculated by unit 840 based onsquared minima metrics from equation (22) as:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {\log \left\lbrack \frac{\max\limits_{{x:b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\max\limits_{{x:b_{k}} = 1}{\exp \left( {- {d(x)}} \right)}} \right\rbrack}} \\{= {{\min\limits_{{x:b_{k}} = 1}{d(x)}} - {\min\limits_{{x:b_{k}} = 0}{d(x)}}}} \\{= {\log \left\lbrack \frac{\max\limits_{{x:b_{k}} = 0}{\exp \left( {- \left( {d_{l_{2}}(x)} \right)^{2}} \right)}}{\max\limits_{{x:b_{k}} = 1}{\exp \left( {- \left( {d_{l_{2}}(x)} \right)^{2}} \right)}} \right\rbrack}} \\{= {\left( {\min\limits_{{x:b_{k}} = 1}{d_{l_{2}}(x)}} \right)^{2} - \left( {\min\limits_{{x:b_{k}} = 0}{d_{l_{2}}(x)}} \right)^{2}}}\end{matrix} & (25)\end{matrix}$

It can be observed from equation (25) that un-squared l₂ norms may becomputed during the metric calculation stage of the PC-ML detectionalgorithm. If the variance of effective noise at the receiver is unitary(after pre-whitening, for example), the c^(th) metric d_(l) ₂ _(,c) frommetric calculation block may be computed as:

d_(l) ₂ _(,c)=l₂=∥v∥₂   (26)

-   -   where, v=y−Hx, c=1, 2, . . . , M^(N) ^(t) )

It can be observed from equation (25) that the Max-Log-MAP ML detectionwith post-squaring compensation may utilize only two squared l₂ normsper coded transmission bit representing minimum metric for bit “1”hypothesis and minimum metric for bit “0” hypothesis, instead of M^(N)^(t) squared l₂ norms. Therefore, complexity overhead due to squareoperations can be small, and it may not affect the overall computationalcomplexity. At 750, calculated LLRs for all N_(T)·B coded bitstransmitted over a plurality of spatial sub-channels for a singlefrequency bin may be passed to the outer channel decoder 850 thatprovides decoded data at its output.

The post-squaring compensation may be also applied for the ML detectionwith QR preprocessing (hereinafter abbreviated as PC-QRML detection).FIG. 9 shows a process of Max-Log-MAP QRML detection with post-squaringcompensation, and FIG. 10 shows a block diagram of Max-Log-MAP MLdetector with post-squaring compensation that performs processing flowfrom FIG. 9.

At 910, for every spatial data stream j=1, 2, . . . , N_(T), the channelmatrix may be permuted by unit 1010 such that the rightmost column ofthe permuted matrix corresponds to the j^(th) decoded stream as shown inequation (15). The QR decomposition of permuted channel matrix for everyutilized spatial data stream is then performed, and the received symbolvector may be rotated as defined by equation (17).

At 920, for every spatial data stream j=1, 2, . . . , N_(T), rotatedreceived symbols may be sliced in unit 1020 in order to obtain estimatedmodulation symbols for that particular spatial data stream j. All other(N_(T)−1) spatial data streams may be hypothesized with all possibletransmission symbols. Therefore, unit 1020 calculates N_(T)·M^(N) ^(T)⁻¹ metrics based on l₂ norms. At 930, for every transmission bit k=1, 2,. . . , N_(T)·B, unit 1030 performs search for minima metrics for allhypotheses x for which bit k is equal to bit “0”, and for all hypothesesx for which bit k is equal to bit “1”. Computational complexity of thecandidate search process is therefore proportional toO(N_(T)·B·N_(T)·M^(N) ^(T) ⁻¹).

At 940, once the minima metrics for all coded bits are determined, unit1040 in FIG. 10 may perform the post-squaring compensation of minimametrics. As shown before, the post-squaring of minima metrics in thePC-QRML detection algorithm produces identical results as minima metricsof the original ML detection, i.e. l₂ ² norms are equivalent topost-squared l₂ norms. At 950, for every coded transmission bit k=1, 2,. . . , N_(T)·B, bit LLR L (b_(k)) may be calculated based on squaredminima metrics in unit 1050 as follows:

$\begin{matrix}\begin{matrix}{{L\left( b_{k} \right)} = {\log \left\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- {d(x)}} \right)}}{\max\limits_{{x\text{:}b_{k}} = 1}{\exp \left( {- {d(x)}} \right)}} \right\rbrack}} \\{= {{\min\limits_{{x\text{:}b_{k}} = 1}{d(x)}} - {\min\limits_{{x\text{:}b_{k}} = 0}{d(x)}}}} \\{= {\log \left\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}{\exp \left( {- \left( {d_{l_{2}}(x)} \right)^{2}} \right)}}{\max\limits_{{x\text{:}b_{k}} = 1}{\exp \left( {- \left( {d_{l_{2}}(x)} \right)^{2}} \right)}} \right\rbrack}} \\{= {\left( {\min\limits_{{x\text{:}b_{k}} = 1}{d_{l_{2}}(x)}} \right)^{2} - \left( {\min\limits_{{x\text{:}b_{k}} = 0}{d_{l_{2}}(x)}} \right)^{2}}}\end{matrix} & (27)\end{matrix}$

Assuming unitary variance of the effective noise at the receiver (afterpre-whitening, for example), the c^(th) metric d_(l) ₂ _(,c) from themetric calculation block 1020 may be determined as:

d_(l) ₂ _(,c)=l₂=∥v∥₂   (28)

-   -   where, v=y_(r)−Rx, c=1, 2, . . . , N_(t)M^(N) ^(t) ⁻¹

At 960, calculated LLRs for all N_(T)·B coded bits transmitted over aplurality of spatial sub-channels for a single frequency bin may bepassed to the outer channel decoder 1060 that provides decoded bits atits output.

The calculation of un-squared l₂ norm may require smaller dynamic rangethan the calculation of l₂ ² norm caused by range compression of thesquare-root operations from equation (8). Therefore, a smallerarithmetic precision for representation of l₂ norms may be required thanfor representation of l₂ ² norms. This leads to smaller memoryrequirements for PC-ML and PC-QRML detection algorithms.

Another advantage of the post-squaring compensation may be a reductionin complexity for calculation of metrics. While it may appear that theML detection with the post-squaring compensation may require morecomputations than the original approach based on l₂ ² norm due tosquare-root operations from equation (8). But, this holds only if theexact un-squared l₂ norm is calculated. However, computationalcomplexity of the PC-QRML detection may be significantly reduced if thecalculation of l₂ norm is approximated.

There are two aspects for approximate calculation of un-squared l₂ normsthat may need to be considered: computational complexity and accuracy.By applying the “maxsum” approximation, according to the equation below,computation of l₂ norms may be accurately obtained with the summationand comparison operations instead of square-root operations:

$\begin{matrix}{\begin{matrix}{\mspace{79mu} {d_{l_{2},c} = {{v}_{2} = \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}}}}} \\{\approx {{\max \left( {z_{j}} \right)} + {F \cdot \left( {\sum\limits_{j,{j \neq m}}{z_{j}}} \right)}}} \\{= {{\max \left( {z_{j}} \right)} + {F \cdot \left( {\left( {\sum\limits_{j}{z_{j}}} \right) - {\max \left( {z_{j}} \right)}} \right)}}}\end{matrix}\mspace{79mu} {where}\begin{matrix}{z = \left\lbrack {{{real}\left( v_{1} \right)},{{imag}\left( v_{1} \right)},{{real}\left( v_{2} \right)},{{imag}\left( v_{2} \right)},\ldots \mspace{11mu},{{real}\left( v_{N_{t}} \right)},{{imag}\left( v_{N_{t}} \right)}} \right\rbrack^{T}} \\{= \left\lbrack {z_{1},z_{2},\ldots \mspace{11mu},z_{2\; N_{t}}} \right\rbrack^{T}}\end{matrix}} & (29) \\{\mspace{79mu} {m = {\underset{j}{\arg \; \max}\left( {z_{j}} \right)}}} & \;\end{matrix}$

Parameter F is a tuning factor that may be determined by simulations.

FIG. 11 shows comparison of computational complexity between several MLdetection schemes. The number of utilized arithmetic operations indifferent algorithms is compared. In this exemplary case, thebit-precision for representation of LLRs and memory requirements are notconsidered as factors for comparison of computational complexity. As anillustrative example, it may be assumed a wireless system with twotransmit and two receive antennas, 64-QAM modulation may be applied atthe transmitter, the Max-Log-MAP ML detection and QRML detection mayutilize squared l₂ norms for computation of metric. The PC-ML detectionand the PC-QRML detection may utilize “maxsum” approximation forcalculation of un-squared l₂ norms given by equation (29) with thetuning factor F=0.25.

The number of arithmetic operations provided in FIG. 11 corresponds tothe detection of two spatial data streams for a single frequencysub-band with 12 coded transmission bits that represent two 64-QAMmodulation symbols. It can be observed that ML and QRML algorithms uselarge number of real multiplications to calculate metrics. On the otherside, in the PC-ML and PC-QRML algorithms, real multiplications may bereplaced by comparison and select operations. Also, computationalcomplexity of minima search may be reduced. The post-squaringcompensation assumes only small number of multiplications: two realmultiplications per coded transmission bit. Therefore, totalcomputational complexity of detection algorithms with post-squaringcompensation may be substantially smaller.

Simulations have been conducted in order to evaluate error rateperformance of different detection algorithms. All analyzed schemesutilize exact calculation of norms and floating point representation ofnumbers. A wireless system with two transmit and two receive antennasmay be assumed with Vertical encoding and with 64-QAM modulation typemay be applied at the transmitter. Two different coding schemes may beutilized in simulations: tailbiting convolutional codes (TBCC) withrates ½, ⅔ and ¾, and convolution Turbo codes (CTC) with rates ½, ⅔, 3/4and ⅚. The number of data packets used in simulations is equal to 10000,it is assumed perfect channel state information at the receiver. Thefading scenario is Vehicular-A channel with the speed of mobilesubscriber of 30 km/h, which corresponds to the Doppler frequency of 64Hz. The carrier frequency of 2.3 GHz may be utilized.

FIG. 12 summarizes features of ML detection schemes that are evaluatedby aforementioned simulations. It can be observed that l₂ and squared l₂norms may be exploited for metric calculation in different detectionalgorithms. The reference algorithm in terms of error-rate performanceis the QRML detection with metric based on squared l₂ norms because ofoptimal error rate performance.

FIG. 13 illustrates loss in dB units of QRML detection with metric basedon non-squared l₂ norms and of PC-QRML detection, relative to the QRMLdetection with metric based on direct computation of squared l₂ norms atthe packet error rate (PER) of 10⁻². It can be observed that the PC-QRMLdetection algorithm shows identical PER performance as the QRMLdetection with metric based on direct computation of squared l₂ norms.Also, the QRML detection with non-squared norms (post-squaring is notapplied) may experience error-rate performance degradation especially inthe case of CTC codes.

The various operations of methods described above may be performed byvarious hardware and/or software component(s) and/or module(s)corresponding to means-plus-function blocks illustrated in the Figures.Generally, where there are methods illustrated in Figures havingcorresponding counterpart means-plus-function Figures, the operationblocks correspond to means-plus-function blocks with similar numbering.For example, blocks 710-750 illustrated in FIG. 7 correspond tomeans-plus-function blocks 710A-750A illustrated in FIG. 7A. Similarly,blocks 910-960 illustrated in FIG. 9 correspond to means-plus-functionblocks 910A-960A illustrated in FIG. 9A.

The various illustrative logical blocks, modules and circuits describedin connection with the present disclosure may be implemented orperformed with a general purpose processor, a digital signal processor(DSP), an application specific integrated circuit (ASIC), a fieldprogrammable gate array signal (FPGA) or other programmable logic device(PLD), discrete gate or transistor logic, discrete hardware componentsor any combination thereof designed to perform the functions describedherein. A general purpose processor may be a microprocessor, but in thealternative, the processor may be any commercially available processor,controller, microcontroller or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

The steps of a method or algorithm described in connection with thepresent disclosure may be embodied directly in hardware, in a softwaremodule executed by a processor, or in a combination of the two. Asoftware module may reside in any form of storage medium that is knownin the art. Some examples of storage media that may be used includerandom access memory (RAM), read only memory (ROM), flash memory, EPROMmemory, EEPROM memory, registers, a hard disk, a removable disk, aCD-ROM and so forth. A software module may comprise a singleinstruction, or many instructions, and may be distributed over severaldifferent code segments, among different programs, and across multiplestorage media. A storage medium may be coupled to a processor such thatthe processor can read information from, and write information to, thestorage medium. In the alternative, the storage medium may be integralto the processor.

The methods disclosed herein comprise one or more steps or actions forachieving the described method. The method steps and/or actions may beinterchanged with one another without departing from the scope of theclaims. In other words, unless a specific order of steps or actions isspecified, the order and/or use of specific steps and/or actions may bemodified without departing from the scope of the claims.

The functions described may be implemented in hardware, software,firmware or any combination thereof. If implemented in software, thefunctions may be stored as one or more instructions on acomputer-readable medium. A storage media may be any available mediathat can be accessed by a computer. By way of example, and notlimitation, such computer-readable media can comprise RAM, ROM, EEPROM,CD-ROM or other optical disk storage, magnetic disk storage or othermagnetic storage devices, or any other medium that can be used to carryor store desired program code in the form of instructions or datastructures and that can be accessed by a computer. Disk and disc, asused herein, include compact disc (CD), laser disc, optical disc,digital versatile disc (DVD), floppy disk, and Blu-ray® disc where disksusually reproduce data magnetically, while discs reproduce dataoptically with lasers.

Software or instructions may also be transmitted over a transmissionmedium. For example, if the software is transmitted from a website,server, or other remote source using a coaxial cable, fiber optic cable,twisted pair, digital subscriber line (DSL), or wireless technologiessuch as infrared, radio, and microwave, then the coaxial cable, fiberoptic cable, twisted pair, DSL, or wireless technologies such asinfrared, radio, and microwave are included in the definition oftransmission medium.

Further, it should be appreciated that modules and/or other appropriatemeans for performing the methods and techniques described herein can bedownloaded and/or otherwise obtained by a user terminal and/or basestation as applicable. For example, such a device can be coupled to aserver to facilitate the transfer of means for performing the methodsdescribed herein. Alternatively, various methods described herein can beprovided via storage means (e.g., RAM, ROM, a physical storage mediumsuch as a compact disc (CD) or floppy disk, etc.), such that a userterminal and/or base station can obtain the various methods uponcoupling or providing the storage means to the device. Moreover, anyother suitable technique for providing the methods and techniquesdescribed herein to a device can be utilized.

It is to be understood that the claims are not limited to the preciseconfiguration and components illustrated above. Various modifications,changes and variations may be made in the arrangement, operation anddetails of the methods and apparatus described above without departingfrom the scope of the claims.

1. A method for detection of data transmitted on a plurality of spatialchannels in a multiple-input multiple-output wireless communicationssystem, comprising: calculating un-squared l₂ norm metrics for a set ofhypothesized symbols that correspond to coded bits transmitted over thespatial channels; searching the un-squared l₂ norm metrics for optimummetrics; squaring optimum metrics found during the searching to obtainpost-squared metrics; and calculating log-likelihood ratio values forcoded bits transmitted over the spatial channels using post-squaredmetrics.
 2. The method of claim 1, comprising: decoding bits transmittedover the plurality of spatial sub-channels by using the calculatedlog-likelihood ratios.
 3. The method of claim 1, wherein searching theun-squared l₂ norm metrics for optimum metrics comprises: searching theun-squared l₂ norm metrics for minima.
 4. The method of claim 1, whereincalculating un-squared l₂ norm metrics for a set of hypothesized symbolsthat correspond to coded bits transmitted over the spatial channelscomprises: utilizing approximated l₂ norm values rather than exact l₂norm values.
 5. The method of claim 1, further comprising: permuting amatrix of channel estimates for the spatial channels, to generatepermuted matrices of channel estimates for the plurality of spatialstreams; and performing QR decomposition of the permuted matrices togenerate a unitary matrix and an upper triangular matrix for theplurality of spatial streams.
 6. The method of claim 5, furthercomprising: rotating a received signal with the unitary matrices togenerate filtered outputs; slicing filtered outputs to obtain slicedcoded bits for the plurality of spatial streams; hypothesizing symbolsthat may be transmitted on the spatial streams using sliced coded bits;and decoding bits transmitted over the plurality of spatial sub-channelsby using calculated log-likelihood ratios of coded bits.
 7. An apparatusfor detection of data transmitted on a plurality of spatial channels ina multiple-input multiple-output wireless communications system,comprising: logic for calculating un-squared l₂ norm metrics for a setof hypothesized symbols that correspond to coded bits transmitted overthe spatial channels; logic for searching the un-squared l₂ norm metricsfor optimum metrics; logic for squaring optimum metrics found during thesearching to obtain post-squared metrics; and logic for calculatinglog-likelihood ratio values for coded bits transmitted over the spatialchannels using post-squared metrics.
 8. The apparatus of claim 7,comprising: logic for decoding bits transmitted over the plurality ofspatial sub-channels by using the calculated log-likelihood ratios. 9.The apparatus of claim 7, wherein the logic for searching the un-squaredl₂ norm metrics for optimum metrics is configured to search theun-squared l₂ norm metrics for minima.
 10. The apparatus of claim 7,wherein the logic for calculating un-squared l₂ norm metrics for a setof hypothesized symbols that correspond to coded bits transmitted overthe spatial channels is configured to utilize approximated l₂ normvalues rather than exact l₂ norm values.
 11. The apparatus of claim 7,further comprising: logic for permuting a matrix of channel estimatesfor the spatial channels, to generate permuted matrices of channelestimates for the plurality of spatial streams; and logic for performingQR decomposition of the permuted matrices to generate a unitary matrixand an upper triangular matrix for the plurality of spatial streams. 12.The apparatus of claim 11, further comprising: logic for rotating areceived signal with the unitary matrices to generate filtered outputs;logic for slicing filtered outputs to obtain sliced coded bits for theplurality of spatial streams; logic for hypothesizing symbols that maybe transmitted on the spatial streams using sliced coded bits; and logicfor decoding bits transmitted over the plurality of spatial sub-channelsby using calculated log-likelihood ratios of coded bits.
 13. Anapparatus for detection of data transmitted on a plurality of spatialchannels in a multiple-input multiple-output wireless communicationssystem, comprising: means for calculating un-squared l₂ norm metrics fora set of hypothesized symbols that correspond to coded bits transmittedover the spatial channels; means for searching the un-squared l₂ normmetrics for optimum metrics; means for squaring optimum metrics foundduring the searching to obtain post-squared metrics; and means forcalculating log-likelihood ratio values for coded bits transmitted overthe spatial channels using post-squared metrics.
 14. The apparatus ofclaim 13, comprising: means for decoding bits transmitted over theplurality of spatial sub-channels by using the calculated log-likelihoodratios.
 15. The apparatus of claim 13, wherein the means for searchingthe un-squared l₂ norm metrics for optimum metrics is configured tosearch the un-squared l₂ norm metrics for minima.
 16. The apparatus ofclaim 13, wherein the means for calculating un-squared l₂ norm metricsfor a set of hypothesized symbols that correspond to coded bitstransmitted over the spatial channels is configured to utilizeapproximated l₂ norm values rather than exact l₂ norm values.
 17. Theapparatus of claim 13, further comprising: means for permuting a matrixof channel estimates for the spatial channels, to generate permutedmatrices of channel estimates for the plurality of spatial streams; andmeans for performing QR decomposition of the permuted matrices togenerate a unitary matrix and an upper triangular matrix for theplurality of spatial streams.
 18. The apparatus of claim 17, furthercomprising: means for rotating a received signal with the unitarymatrices to generate filtered outputs; means for slicing filteredoutputs to obtain sliced coded bits for the plurality of spatialstreams; means for hypothesizing symbols that may be transmitted on thespatial streams using sliced coded bits; and means for decoding bitstransmitted over the plurality of spatial sub-channels by usingcalculated log-likelihood ratios of coded bits.
 19. A computer-programproduct for detection of data transmitted on a plurality of spatialchannels in a multiple-input multiple-output wireless communicationssystem, comprising a computer readable medium having instructions storedthereon, the instructions being executable by one or more processors andthe instructions comprising: instructions for calculating un-squared l₂norm metrics for a set of hypothesized symbols that correspond to codedbits transmitted over the spatial channels; instructions for searchingthe un-squared l₂ norm metrics for optimum metrics; instructions forsquaring optimum metrics found during the searching to obtainpost-squared metrics; and instructions for calculating log-likelihoodratio values for coded bits transmitted over the spatial channels usingpost-squared metrics.
 20. The computer-program product of claim 19,wherein the instructions further comprise: instructions for decodingbits transmitted over the plurality of spatial sub-channels by using thecalculated log-likelihood ratios.
 21. The computer-program product ofclaim 19, wherein the instructions for searching the un-squared l₂ normmetrics for optimum metrics comprise instructions for searching theun-squared l₂ norm metrics for minima.
 22. The computer-program productof claim 19, wherein the instructions for calculating un-squared l₂ normmetrics for a set of hypothesized symbols that correspond to coded bitstransmitted over the spatial channels further comprise instructions forutilizing approximated l₂ norm values rather than exact l₂ norm values.23. The computer-program product of claim 19, wherein the instructionsfurther comprise: instructions for permuting a matrix of channelestimates for the spatial channels, to generate permuted matrices ofchannel estimates for the plurality of spatial streams; and instructionsfor performing QR decomposition of the permuted matrices to generate aunitary matrix and an upper triangular matrix for the plurality ofspatial streams.
 24. The computer-program product of claim 23, whereinthe instructions further comprise: instructions for rotating a receivedsignal with the unitary matrices to generate filtered outputs;instructions for slicing filtered outputs to obtain sliced coded bitsfor the plurality of spatial streams; instructions for hypothesizingsymbols that may be transmitted on the spatial streams using slicedcoded bits; and instructions for decoding bits transmitted over theplurality of spatial sub-channels by using calculated log-likelihoodratios of coded bits.